Using partial differential equations to generate free-form surfaces

doi:10.1016/0010-4485(90)90049-I

Computer-Aided Design

Volume 22, Issue 4, May 1990, Pages 202-212

https://doi.org/10.1016/0010-4485(90)90049-I Get rights and content

Abstract

A method of generating free-form surfaces using solutions to a suitably chosen partial differential equation is discussed. By varying the boundary conditions and a parameter in the partial differential equation, it is demonstrated that a wide variety of surface shapes are accessible to the method.

References (15)

T. Várady
An experimental system for interactive design and manufacture of sculptured surfaces
Comput. Ind.
(1982)
L. Piegl et al.
Curve and surface constructions using rational B-splines
Comput.-Aided Des.
(1987)
C.D. Woodward
Cross-sectional design of B-spline surfaces
Comput. Graph
(1987)
C.D. Woodward
Skinning techniques for interactive B-spline surface interpolation
Comput.-Aided Des.
(1988)
M.I.G. Bloor et al.
Generating blend surfaces using partial differential equations
Comput.-Aided Des.
(1989)
I.D. Faux et al.
Computational Geometry for Design and Manufacture
(1979)
M.E. Mortenson
Geometric Modeling
(1985)

There are more references available in the full text version of this article.

Cited by (190)

Surface Blending Using T-splines in Semi-NURBS Form
2022, CAD Computer Aided Design
Citation Excerpt :
Especially in the aeroengine design, a high-quality blending surface between the blade and shroud is of significance to a high-performance engine [1]. Due to the usefulness of surface blending, many classical techniques have emerged, such as the rolling-ball method [2,3], the PDE-based method [4,5], and the linear combination method [6,7]. With the development of numerical computing technologies like isogeometric analysis (IGA) [8], the prospect for a new generation of integrated design environment is becoming clear [9], producing some new requirements that have not been fully concerned by traditional surface blending approaches.
Surface blending is a permanent topic in computer aided geometric design. This modeling operation that generates smooth connection between given surfaces has been studied by many researchers. However, as the interests grow of integrated modeling system based on isogeometric analysis (IGA), some requirements that have not been fully discussed by conventional approaches become crucial, such as the exact spline representation of the blending surface, and the seamless merging of the modeling results. To address these problems, a surface blending method based on T-splines in semi-NURBS form is proposed in this paper. The flexibility and unstructured nature of T-splines provide a powerful foundation to achieve blending construction between incompatible NURBS or T-spline surfaces without disturbing them or inserting any redundant control points. Through a reasonable arrangement of the topological T-mesh, $C^{2}$ continuity constraints between T-splines are transformed into explicit formulae to construct the blending surface. Additionally, the generated model is one single watertight T-spline surface that can be easily used for isogeometric analysis. Several experimental examples validate the effectiveness and flexibility of the proposed blending method.
Construction of freeform mirrors for an off-axis telecentric scanning system through multiple surfaces expansion and mixing
2020, Results in Physics
In this paper, a direct design method for an off-axis two-mirror telecentric scanning system with a linear field of view (FOV) is proposed. A single freeform mirror structure is firstly considered, in which the aberration free geometry of the off-axis parabolic (OAP) surface is leveraged to provide the focusing function and build surface contour of the sub-region on the mirror for each FOV. Multiple OAP surfaces for construction of the freeform mirror are located at an OAP base to satisfy the telecentric condition. The imaging distortion of this single freeform mirror structure is analyzed and found unavoidable due to the unsymmetrical geometry of the OAP base. A freeform reflective corrector is supplemented, and it is constructed from multiple plane surfaces located at a curved base to fulfill the f-theta scanning geometry. Thus, a two-mirror structure composed of one freeform primary mirror and one freeform reflective corrector is established. Each plane-OAP surfaces pair corresponds to a specific FOV. These multiple OAP surfaces and multiple plane surfaces are then expanded and mixed respectively, to construct the freeform primary mirror and freeform reflective corrector. An f-theta two-mirror freeform scanning system with ±10.4° linear FOV is designed using the proposed construction method. The design result is diffraction-limited, and a scanning error less than 5 μm and telecentricity angle less than 0.2° are achieved.
Construction of triharmonic Bézier surfaces from boundary conditions
2020, Journal of Computational and Applied Mathematics
The surface of partial differential equation (PDE surface) is a surface that satisfies the PDE with boundary conditions, which can be applied in surface modeling and construction. In this paper, the construction of tensor product Bézier surfaces of triharmonic equation from different boundary conditions is presented. The internal control points of the resulting triharmonic Bézier surface can be obtained uniquely by the given boundary condition. Some representative examples show the effectiveness of the presented method.
Shape transformation using the modified Allen–Cahn equation
2020, Applied Mathematics Letters
In this paper, we propose a phase-field model for the shape transformation and its simple numerical method. The proposed phase-field model is based on the Allen–Cahn equation, which is a second-order nonlinear parabolic partial differential equation and originally arises from modeling phase separation in alloys. The numerical scheme is a hybrid method using the operator splitting method. To validate the proposed phase-field model for the shape transformation, we perform two- and three-dimensional shape metamorphosis. The proposed phase-field model is also compared with the linear interpolation. The computational results demonstrate that the proposed mathematical model satisfactorily and naturally simulate the shape transformation.
Bicubic B-spline surfaces constrained by the Biharmonic PDE
2019, Applied Mathematics and Computation
Bicubic B-spline surface constrained by the Biharmonic PDE is presented in this paper. By representing the Biharmonic PDE in the form of the bilinear B-spline bases, we find the regular vector-valued coefficients and discover that bicubic B-spline surface can satisfy the Biharmonic PDE. When the control points of the boundaries for open or closed surfaces are given, the inner control points can be fully determined. For each case of the surfaces open in both directions, closed in one direction and closed in both directions, a linear system for solving inner control points is established. Some examples show the effectiveness of the given method.
Explicit Bézier control net of a PDE surface
2017, Computers and Mathematics with Applications
The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface.
In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.

View all citing articles on Scopus

View full text

Using partial differential equations to generate free-form surfaces

Abstract

Comput. Ind.

Comput.-Aided Des.

Comput. Graph

Comput.-Aided Des.

Comput.-Aided Des.

Computational Geometry for Design and Manufacture

Geometric Modeling